QUANTUM COHOMOLOGY OF LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS
HARRY TAMVAKIS
Abstract. This is the written version of my talk at the Mathematische Ar- beitstagung 2001, which reported on joint work with Andrew Kresch (papers [KT2] and [KT3]).
The Classical Theory
For the purposes of this talk we will work over the complex numbers, and in the holomorphic (or algebraic) category. There are three kinds of Grassmannians that we will consider, depending on their Lie type:
• In type A,letG(k, N)=SLN /Pk be the Grassmannian which parametrizes N k-dimensional linear subspaces of C .
• In type C,letLG = LG(n, 2n)=Sp2n/Pn be the Lagrangian Grassmannian.
• In types B and D,letOG = OG(n, 2n +1)=SO2n+1/Pn denote the odd or- thogonal Grassmannian, which is isomorphic to the even orthogonal Grassmannian OG(n +1, 2n +2)=SO2n+2/Pn+1.
We begin by defining the Lagrangian Grassmannian LG(n, 2n), and will discuss the Grassmannians for the other types later in the talk. Let V be the vector 2n V space C , equipped with a symplectic form. A subspace Σ of is isotropic if the restriction of the form to Σ vanishes. The maximal possible dimension of an isotropic subspace is n, and in this case Σ is called a Lagrangian subspace. The Lagrangian Grassmannian LG = LG(n, 2n) is the projective complex manifold which parametrizes Lagrangian subspaces Σ in V . There is a stratification of LG by Schubert varieties Xλ, one for each strict n ` ` λ partition λ =(λ1 >λ2 > ··· >λ` > 0) with λ1 6 . The number = ( )isthe n length of λ, and we let Dn denote the set of strict partitions λ with λ1 6 .To describe Xλ more precisely, fix an isotropic flag of subspaces Fi of V :
0 ⊂ F1 ⊂ F2 ⊂···⊂Fn ⊂ V
with dim(Fi)=i for each i,andFn Lagrangian. Then
i 6 i 6 ` }. Xλ = { Σ ∈ LG | dim(Σ ∩ Fn+1−λi ) > for 1 P The variety Xλ has codimension |λ| := λi in LG, and determines a Schubert class 2|λ| σ ,...,σ H2 LG σλ =[Xλ]inH (LG, Z). The classes 1 n are called special,and ( )
Date: June 16, 2001. 1 2 HARRY TAMVAKIS has rank 1, generated by σ1. We have an isomorphism of abelian groups X
∗
Z · σ H (LG, Z)= λ
λ∈Dn which describes the additive structure of the cohomology H∗(LG). As all cohomol- ogy classes occur in even degrees, we adopt the convention that a class α in the cohomology of a complex manifold X has degree k when α lies in H2k(X). Classically, there are three ingredients necessary for an understanding of the multiplicative structure of H∗(LG). a) A presentation of H∗(LG) in terms of generators and relations. This is due to Borel [Bo], and can be derived as follows. Consider the universal exact sequence of vector bundles over LG ∗ 0 −→ E −→ VLG −→ E −→ 0.
Here VLG is the trivial rank 2n vector bundle over LG, E is the tautological sub- ∗ bundle of VLG, and we have identified the quotient bundle VLG/E with E by using the symplectic form on V . The special Schubert class σi is equal to the i-th Chern
∗
i 6 n LG class ci(E ), for 1 6 . The cohomology ring of can then be presented as a quotient of the polynomial ring in the Chern classes of E by the relations coming from the Whitney sum formula ∗ c(E)c(E )=(1− σ1 + σ2 −···)(1 + σ1 + σ2 + ···)=1. ∗ σ ,...,σ Expanding this equation, one obtains that H (LG) is a quotient of Z[ 1 n] by the relations Xn−i 2 k σi +2 (−1) σi+kσi−k =0
k=1
i 6 n σ j< for 1 6 , where it is understood that j =0if 0. b) A Giambelli formula which identifies the polynomials which represent the Schu- bert classes σλ in the above presentation. For the type A Grassmannians G(k, N), the answer to this question is quite classical (and due to Giambelli). In contrast, an analogous formula for the Lagrangian Grassmannian was proved much more re- cently. According to Pragacz [P], the Giambelli formula for LG is obtained in two stages: for i>j>0, we have Xn−i k σi,j = σiσj +2 (−1) σi+kσj−k k=1 while for any λ ∈Dn of length at least 3,
σ σ .
i
Quantum Cohomology of LG(n, 2n)
We will work throughout with the small quantum cohomology ring QH∗(LG). This is a deformation of H∗(LG) which first occured in the work of string theorists. The multiplicative structure of the quantum cohomology ring of LG encodes the enumerative geometry of rational curves in LG, in the form of Gromov–Witten invariants. ∗ q q The ring QH (LG) is an algebra over Z[ ], where is a formal variable of degree n + 1; one recovers the classical cohomology ring by setting q = 0. To describe the product structure, recall that for each Schubert class σλ one has a (Poincar´e) dual class σλ0 , defined so that Z
σλ σµ = δλµ0 LG 0 for all partitions λ, µ in Dn. The parts of the partition λ complement the parts of λ in the set {1,...,n}. Using dual classes, we can write the classical structure ν constant eλµ in (1) as a triple intersection number: Z ν eλµ = hσλ,σµ,σν0 i0 = σλ σµ σν0 . LG In QH∗(LG) there is a formula X d (2) σλ · σµ = hσλ,σµ,σν0 id σν q , ν |ν| |λ| |µ|−d n the sum over d > 0 and strict partitions with = + ( + 1). The Gromov–Witten invariant hσλ,σµ,σν0 id is a non-negative integer, which counts the 1 → LG f ∈ X f ∈ X number of degree-d rational maps f : È such that (0) λ, (1) µ and f(∞) ∈ Xν0 , for general translates of the three Schubert varieties Xλ, Xµ 1 0 dσ 0 and Xν .Heredegree d means that f∗[È ]= 1 . The ‘miracle’ is that with this definition, (2) gives an associative product. There is a long list of papers studying questions (a), (b) and (c) above for the quantum cohomology rings of homogeneous spaces. We now have answers to all
three questions when X = SLN /P is a partial flag variety of SLN (C ); see [W], [ST], [Be] and [BCF] for the case of the Grassmannian G(k, N), and [KT2] for more references. For an arbitrary complex semisimple Lie group G, with Borel subgroup B, Kim [K] found a presentation of QH∗(G/B), thus answering (a). However, one still lacked a presentation of the quantum ring for the other parabolic subgroups and a quantum Giambelli formula to compute the Gromov–Witten invariants, if the Lie group G is not of type A. This was due to serious difficulties in extending the methods used in the proofs in the SLN case to the other Lie types (for a discussion of one such problem, see [KT1, §4]). The remainder of this talk reports on joint work with Andrew Kresch [KT1] [KT2] [KT3]. We answer all three basic questions for the quantum cohomology rings 4 HARRY TAMVAKIS of the type B, C and D Grassmannians mentioned previously. For the Lagrangian Grassmannian, questions (a) and (b) are addressed by the following Theorem. Theorem 1. The ring QH∗(LG) is presented as a quotient of the polynomial ring σ ,...,σ ,q Z[ 1 n ] by the relations Xn−i 2 k n−i σi +2 (−1) σi+kσi−k =(−1) σ2i−n−1 q. k=1
The Schubert class σλ in this presentation is given by the Giambelli formulas Xn−i k n+1−i σi,j = σiσj +2 (−1) σi+kσj−k +(−1) σi+j−n−1 q k=1 for i>j>0,and
σ σ ,
i Note that Theorem 1 is not saying that the cohomology ring of LG(n+1, 2n+2) ∗ 2 is isomorphic to QH (LG(n, 2n)), for the relation σn+1 = 0 in the former ring fails to hold in the latter (as q2 =6 0). It turns out that the algebra of Schur Q- polynomials is not sufficient to describe the multiplication in QH∗(LG). Instead, we use Qe-polynomials; this is a family of symmetric polynomials modelled on Schur’s Q- polynomials, and defined by Pragacz and Ratajski [PR] in their work on degeneracy loci. The Qe-polynomials seem to be the correct geometric analogue of Schur’s S- polynomials in type C. Their multiplication also governs the Arakelov theory of LG, regarded as a scheme over the ring of rational integers [T]. One consequence of Theorem 1 is a quantum Pieri rule for multiplying a class ∗ σλ in H (LG) by a special Schubert class, which extends the classical one of [BH]. For another, define an involution ∗ on Dn by ∗ (λ1, ..., λ`)=(n +1− λ`,...,n+1− λ1), and let ρn be the partition (n, n − 1,...,1) (which corresponds to the class of a point in LG). We then have the following rule for multiplying by the point class: `(λ) ∗ n 0∗ Theorem 2. σλ ·σρn = σλ q in QH (LG).Inparticular,σρn ·σρn = q ,which implies that there is a unique rational curve of degree n passing through 3 general points on LG(n, 2n). The above results are part of the quantum Schubert calculus for LG. The Orthogonal Grassmannian OG(n, 2n +1) 0 2n+1 Consider now V = C equipped with a non-degenerate symmetric form. The maximal isotropic subspaces in V 0 are n-dimensional and parametrized by the odd ∗ orthogonal Grassmannian OG = OG(n, 2n + 1). The cohomology ring H (OG, Z) QH∗ OG has an additive basis of Schubert classes τλ,forλ ∈Dn.AsaZ-module, ( ) ∗ q q n is identified with H (OG) ⊗ Z[ ], but this time the degree of is 2 . QUANTUM COHOMOLOGY OF LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 5 Theorem 3. The ring QH∗(OG) is presented as a quotient of the polynomial ring τ ,...,τ ,q Z[ 1 n ] modulo the relations Xi−1 2 k i τi +2 (−1) τi+kτi−k +(−1) τ2i =0 k=1 for all i The Schubert class τλ in this presentation is given by the Giambelli formulas Xj−1 k j τi,j = τiτj +2 (−1) τi+kτj−k +(−1) τi+j k=1 for i>j>0,and τ τ , i dim C ( ) Schubert classes deg( ) OG(n, 2n +1) n(n +1)/2 τλ ,λ∈Dn 2n LG(n − 1, 2n − 2) n(n − 1)/2 σλ ,λ∈Dn−1 n In the next result, ∗ and 0 denote the involutions that were defined previously, but on the set Dn−1. Theorem 4. Suppose that λ ∈Dn−1 is a partition with `(λ)=2d + e +1for some nonnegative integers d and e. For any µ, ν ∈Dn−1, we have an equality (5) hτλ,τµ,τν id = hσλ∗ ,σµ0 ,σν0 ie of Gromov–Witten invariants for OG(n, 2n+1) and LG(n−1, 2n−2), respectively. If λ is zero or `(λ) < 2d +1,thenhτλ,τµ,τν id =0. Remarks. 1) The left hand side of (5) is obviously symmetric in λ, µ and ν, unlike 3 / Z the right hand side. This reflects a (Z 2 ) -symmetry enjoyed by the Gromov– Witten invariants for both LG and OG. 2) We lack a purely geometric argument that would ‘explain’ Theorem 4. Instead, we first prove the main Theorems 1 and 3 and develop the quantum Schubert calculus for both spaces, then deduce Theorem 4 using algebra. 6 HARRY TAMVAKIS Proofs of the Main Results The most difficult part of our approach to Theorems 1 and 3 is to establish the quantum Giambelli formulas (3) and (4). The arguments needed come from two sources: algebraic geometry and combinatorics. In conclusion, we mention three key ingredients that occur in the proofs; for precise definitions of the terms that follow, see [KT1], [KT2] and [KT3]. • Intersection theory on the Lagrangian (resp. orthogonal) Quot scheme which com- 1 LG OG pactifies the moduli space of degree d maps from È to (resp. ). These Quot schemes are singular in general. • Degeneracy loci for isotropic morphisms of vector bundles. • Combinatorics of Pe, Qe and Schubert polynomials for the Lie types B, C and D. References [Be] A. Bertram : Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305. [BCF] A. Bertram, I. Ciocan-Fontanine and W. Fulton : Quantum multiplication of Schur poly- nomials,J.Algebra219 (1999), no. 2, 728–746. [BH] B. Boe and H. Hiller : Pieri formula for SO2n+1/Un and Spn/Un,Adv.inMath.62 (1986), 49–67. [Bo] A. Borel : Sur la cohomologie des espaces fibr´es principaux et des espaces homog`enes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115–207. [K] B. Kim : Quantum cohomology of flag manifolds G/B and quantum Toda lattices, Annals of Math. 149 (1999), 129–148. [KT1] A. Kresch and H. Tamvakis : Double Schubert polynomials and degeneracy loci for the classical groups, preprint (2000), available at http://www.math.upenn.edu/∼ kresch. [KT2] A. Kresch and H. Tamvakis : Quantum cohomology of the Lagrangian Grassmannian, preprint (2001), available at http://www.math.upenn.edu/∼ kresch. [KT3] A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians,preprint (2001), available at http://www.math.upenn.edu/∼ kresch. [P] P. Pragacz : Algebro-geometric applications of Schur S-andQ-polynomials,S´eminare d’Alg`ebre Dubreil-Malliavin 1989-1990, Lecture Notes in Math. 1478 (1991), 130–191, Springer-Verlag, Berlin, 1991. [PR] P. Pragacz and J. Ratajski : Formulas for Lagrangian and orthogonal degeneracy loci; Qe -polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87. [S] I. Schur : Uber¨ die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155–250. [ST] B. Siebert and G. Tian : On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679–695. [St] J. R. Stembridge : Shifted tableaux and the projective representations of symmetric groups, Adv. in Math. 74 (1989), 87–134. [T] H. Tamvakis : Arakelov theory of the Lagrangian Grassmannian, J. reine angew. Math. 516 (1999), 207-223. [W] E. Witten : The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, & physics, 357–422, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995. University of Pennsylvania and Max-Planck-Institut E-mail address: [email protected]